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I've read in multiple quantum mechanics books that the name "continuous" of the continuous spectra is said continuous because in many examples it is an interval of values. But I couldn't find references for this...

Is the continuous spectra always "continuous"? And when the spectra is "continuous", is the continuous part always in the continuous spectra?

Also, can bounded operators have continuous spectra?

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No, the continuous spectrum can also be just a single point, which is the case for the operator $H^{-1}$, where $H$ is the Hamiltonian of the harmonic oscillator, as explained here. More generally, the continuous spectrum of any self-adjoint compact operator is either empty or only contains $0$ as an element.

Bounded operators can have continuous spectra. For example, consider the Hilbert space $L^2([0,1])$ and the position operator $X:L^2([0,1])\longrightarrow L^2([0,1])$ with $(X\psi)(x):=x\psi(x)$. This operator has a purely continuous spectrum but is bounded.

Conversely, unbounded operators can have a discrete spectrum, such as the harmonic oscillator Hamiltonian $H$.

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    $\begingroup$ Continuing this answer a bit: the entire spectrum of a self-adjoint bounded operator can be any bounded, closed subset of the real line. For example, it can be the very-disconnected Cantor middle-thirds set, so containing no intervals at all, though it has cardinality of the continuum, etc. $\endgroup$
    – paul garrett
    Commented Apr 24, 2023 at 14:02

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